NCO_TETRAHEDERON
Newton-Cotes Open Quadrature for the Tetrahedron

NCO_TETRAHEDERON is a C++ library which defines the weights and abscisass for a sequence of 7 Newton-Cotes open quadrature rules for the tetrahedron.

Newton_Cotes rules have the characteristic that the abscissas are equally spaced. For a tetrahedron, this refers to spacing in the unit reference tetrahedron, or in the barycentric coordinate system. These rules may be mapped to an arbitrary tetrahedron, and will still be valid.

The rules are said to be "open" when they do not include points on the boundary of the tetrahedron.

The use of equally spaced abscissas may be important for your application. That may how your data was collected, for instance. On the other hand, the use of equally spaced abscissas carries a few costs. In particular, for a given degree of polynomial accuracy, there will be rules that achieve this accuracy, but use fewer abscissas than Newton-Cotes. Moreover, the Newton-Cotes approach almost always results in negative weights for some abscissas. This is generally an undesirable feature, particularly when higher order quadrature rules are being used.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Related Data and Programs:

CLENSHAW_CURTIS is a C++ library which sets a Clenshaw Curtis quadrature grid in multiple dimensions.

FELIPPA is a C++ library which defines quadrature rules for lines, triangles, quadrilaterals, pyramids, wedges, tetrahedrons and hexahedrons.

GM_RULES is a C++ library which defines a Grundmann-Moeller rule for quadrature over a triangle, tetrahedron, or general M-dimensional simplex.

KEAST is a C++ library which defines a number of quadrature rules for a tetrahedron.

NCC_TETRAHEDRON is a C++ library which defines Newton-Cotes closed quadrature rules on a tetrahedron.

NCO_TETRAHEDRON is available in a C++ version and a FORTRAN90 version and a MATLAB version

NCO_TRIANGLE is a C++ library which defines Newton-Cotes open quadrature rules on a triangle.

NINTLIB is a FORTRAN90 library which contains a variety of routines for numerical estimation of integrals in multiple dimensions.

QUADRATURE_RULES_TET is a dataset directory of triples of files defining various quadrature rules on tetrahedrons.

QUADRULE is a C++ library which defines quadrature rules on a variety of intervals with different weight functions.

STROUD is a C++ library which defines quadrature rules for a variety of unusual areas, surfaces and volumes in 2D, 3D and N-dimensions.

TET_MESH is a C++ library which is useful for work with tetrahedral meshes in 3D;

TETRAHEDRON_MONTE_CARLO, a C++ program which uses the Monte Carlo method to estimate integrals over a tetrahedron.

TETRAHEDRONS, a dataset directory which contains examplesof tetrahedrons;

Reference:

1. Peter Silvester,
   Symmetric Quadrature Formulae for Simplexes,
   Mathematics of Computation,
   Volume 24, Number 109, January 1970, pages 95-100.

Source Code:

• nco_tetrahedron.C, the source code.
• nco_tetrahedron.H, the include file.
• nco_tetrahedron.csh, commands to compile the source code.

Examples and Tests:

• nco_tetrahedron_prb.C, a sample calling program.
• nco_tetrahedron_prb.csh, commands to compile and run the sample program.
• nco_tetrahedron_prb_output.txt, the output file.

List of Routines:

• NCO_TETRAHEDRON_DEGREE: degree of an NCO rule for the tetrahedron.
• NCO_TETRAHEDRON_ORDER_NUM: order of an NCO rule for the tetrahedron.
• NCO_TETRAHEDRON_RULE returns the points and weights of an NCO rule.
• NCO_TETRAHEDRON_RULE_NUM returns the number of NCO rules available.
• NCO_TETRAHEDRON_SUBORDER returns the suborders for an NCO rule.
• NCO_TETRAHEDRON_SUBORDER_NUM returns the number of suborders for an NCO rule.
• NCO_TETRAHEDRON_SUBRULE returns a compressed NCO rule.
• NCO_TETRAHEDRON_SUBRULE_01 returns a compressed NCO rule 1.
• NCO_TETRAHEDRON_SUBRULE_02 returns a compressed NCO rule 2.
• NCO_TETRAHEDRON_SUBRULE_03 returns a compressed NCO rule 3.
• NCO_TETRAHEDRON_SUBRULE_04 returns a compressed NCO rule 4.
• NCO_TETRAHEDRON_SUBRULE_05 returns a compressed NCO rule 5.
• NCO_TETRAHEDRON_SUBRULE_06 returns a compressed NCO rule 6.
• NCO_TETRAHEDRON_SUBRULE_07 returns a compressed NCO rule 7.
• R8MAT_DET_4D computes the determinant of a 4 by 4 R8MAT.
• REFERENCE_TO_PHYSICAL_T4 maps T4 reference points to physical points.
• TETRAHEDRON_VOLUME computes the volume of a tetrahedron.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the C++ source codes.